I cannot see fully where it is attached but it is either a crankcase breather hose or part of the pair valve system. I think it is more likely part of the crankcase breather system. Does the plastic nipple end have an opening in it? If so, is this opening oily? The crankcase breather hose should connect into the back of the air box. At the top of each cylinder, is there another hose coming out of the raised/bump covers that cover the pair valves? Do these hoses join together at a tee and then go into the back of the air box?
No, it will NOT damage the bike. Go on your ride/trip and have a great time.
When attached correctly, it just goes into the back of the airbox. The only drawback to leaving it in the current state is that a tiny amount of oil mist will be blown out of that hose and get onto the top of the engine possibly making a mess. This will NOT increase your oil usage. In the correct configuration, the oil mist just collects inside the airbox and drains into the shorth capped oil collection hose coming out of the bottom of the reservoir.